Space-Time-Frequency Degrees of Freedom: Fundamental Limits for Spatial Information

We bound the number of electromagnetic signals which may be observed over a frequency range [F-W, F+W] a time interval [0, T] within a sphere of radius R. We show that the such constrained signals may be represented by a series expansion whose terms are bounded exponentially to zero beyond a threshold. Our result implies there is a finite amount of information which may be extracted from a region of space via electromagnetic radiation.